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To measure the drag coefficients of spheres over several decades of particle Reynolds number.
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1.0 TITLE
The measurement of drag coefficient of spheres.
2.0 OBJECTIVE
To measure the drag coefficients of spheres over several decades of particle Reynolds
number.
3.0 INTRODUCTION
3.1 Definition of Drag Coefficient for Flow Past Immersed Objects.
The flow of fluids outside bodies appears in many engineering applications
and other processing applications. This occurs, for example, in flow past
spheres in settling, flow through packed beds in drying and filtration, flow past
tubes in heat exchangers and others. It is useful to be able to predict the
frictional losses and/or the force on the submerged objects in these various
applications.
In the examples of fluid friction inside conduits that we considered, the
transfer of momentum perpendicular to the surface resulted in a tangential
shear stress or drag on the smooth surface parallel to the direction flow. This
force exerted by the fluid on the solid in the direction of flow is called skin or
wall drag. For any surface in contact with a flowing fluid, skin friction will
exist. In addition to skin friction, if the fluid is not flowing parallel to the
surface but must change directions to pass around a solid body such as a
sphere, significant additional frictional losses will occur and this is called form
drag.
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3.1.1 Drag Coefficient
In treating fluid flow through pipes and channels, a friction factor, defined as
the ratio of the shear stress to the product of the velocity head and density, was
shown to be used. An analogous factor, called the drag coefficient, is used for
immersed solids. Consider a smooth sphere immersed in a flowing fluid and at
a distance from the solid boundary of the stream sufficient for the approaching
stream to be at a uniform velocity. Define the projected area of the solid body
as the area obtained by projecting the body on a plane perpendicular to the
direction of flow. Denote the projected area by AP. For a sphere, the projected
area is that of a great circle, or ( π4 )D2
p, where DP is the diameter. If FDis the
total drag, the average drag per unit projected area is FD / A p. Just as the
friction factor f is defined as the ratio of τ w to the product of the density of the
fluid and the velocity head, so the drag coefficient CD is defined as the ratio of
FD / A pto this same product or
CD ≡
F D
Ap
ρuO2
2
Where u0 is the velocity of the approaching stream. For particles having
shapes other than spherical, it is necessary to specify the size and geometric
form of the body and its orientation with respect to the direction of flow of the
fluid. One major dimension is chosen as the characteristic length, and the other
important dimensions are given as ratios to the chosen one. For cylinders the
diameter D is taken as the defining dimensions, and the length expressed as
the L/D ratio. The orientation between the particle and the stream is also
specified. For a cylinder, the angle formed by the axis of the cylinder and the
2
direction of flow is sufficient. Then the projected area is can be determined as
LDP, where L is the length of the cylinder. For a cylinder with its axis parallel
to the direction f flow, Ap is( π4 )D 2
p, the same as for sphere of the same
diameter.
From dimension analysis, the drag coefficient of a smooth solid in an
incompressible fluid depends upon a Reynolds number and the necessary
shape ratios. For given shape
CD=Φ(ℜP)
The Reynold’s number for a particle in a fluid is defined as
ℜP ≡G0G p
u
Where D p=characteristic length
G0=uo ρ
A different CD versus ℜP relation exists for each shape and orientation. The
relation must in general be determined experimentally, although a well-
substantiated theoretical equation exists for smooth spheres at low Reynolds
numbers. Drag coefficients for compressible fluids increase with an increase
in the Mach number when the latter becomes greater than about 0.6.
Coefficients in supersonic flow are generally greater than in subsonic flow.
3
4.0 APPARATUS
1. Glass tubes.
2. Drain valves, V1 & V2
3. Knob
4. Ball valves, V3 & V4
4
Glass tubes
Drain valves
Ball valves
Knob
5.0 METHOD
The Instructions Methods
1. Fill the two glass tubes with clear liquids
of different viscosities (cooking oil and
detergent).
The two glass tubes is filled with cooking oil
and another one is filled up with detergent.
2. Mild steel sphere with diameter 3.175mm
are dropped one at a time from the top of the
tubes and allowed to fall to the bottom.
Start the experiment with diameter of
3.175mm by dropping it at one time from top
of the tubes and allowed to fall to bottom.
3. The passage between the 1m marks on the
wall of the tubes being timed with a stop
watch and recorded.
Time is recorded by watching at the passage
between 1m marks on the wall.
4. When each sphere arrives at the recess in
the base of the tubes, it is removed by turning
the valve through 180° by rotating the knob
at the bottom of the glass tube, then open the
ball valve, V3 or V4.
The sphere is removed from the base of the
tubes by rotating the knob at the bottom of
the glass tube, and then only open the ball
valve.
5. After ejecting the sphere, the ball valve
and knobs should be returned to the operating
position.
The sphere is ejected and returned the knob
to the operating position.
6. Repeat the experiment for step 2 until 5
with different dimension and material of the
sphere.
The experiment is repeated by changing the
diameter size of the sphere and the material
of the sphere.
7. Calculate viscosity µ of liquid used. The viscosity of the liquid used is calculated.
5
6.0 RESULTS AND DISCUSSION
RESULTS
Fluid : Cooking Oil
Object shape
MaterialLength/diameter
(mm)Mass (kg)
Time taken (s) Drag coefficient
CD oil
Reynolds No Re oil
Viscosity (kg/sm)1 2 3 Average
Sphere
Mild steel
3.175 0.000142 3.50 3.50 3.60 3.53 4.2559 5.59 0.14605.000 0.000526 1.90 2.00 1.80 1.90 3.8893 2.97 0.80546.350 0.001050 1.40 1.50 1.40 1.43 1.2709 18.74 0.21528.000 0.002040 1.10 1.10 1.10 1.10 0.9338 25.70 0.25709.525 0.003530 0.95 0.95 0.98 0.96 0.8546 27.87 0.3233
Stainless steel
3.175 0.000142 3.81 3.84 3.59 3.75 4.8022 4.96 0.15515.000 0.000525 1.91 1.99 2.00 1.97 4.1811 2.76 0.83516.350 0.001040 1.62 1.58 1.53 1.58 1.5345 15.52 0.23528.000 0.002048 1.12 1.11 1.26 1.16 1.0443 22.98 0.27259.525 0.003527 1.08 0.99 0.99 1.02 0.9648 24.68 0.3435
6
Fluid : Detergent
Object shape
MaterialLength/diameter
(mm)Mass (kg)
Time taken (s) Drag coefficient
CD det
Reynolds No Re det
Viscosity (kg/sm)1 2 3 Average
Sphere
Mild steel
3.175 0.000142 1.00 1.10 1.20 1.10 0.4292 55.45 0.04575.000 0.000526 0.68 0.75 0.72 0.72 0.5800 19.89 0.30656.350 0.001050 0.62 0.60 0.61 0.61 0.2402 99.13 0.09228.000 0.002040 0.50 0.53 0.52 0.52 0.2168 110.72 0.12209.525 0.003530 0.48 0.47 0.49 0.48 0.2220 107.28 0.1624
Stainless steel
3.175 0.000142 0.98 1.10 1.02 1.03 0.3763 63.24 0.04285.000 0.000525 0.83 0.80 0.86 0.83 0.7707 14.97 0.35336.350 0.001040 0.73 0.68 0.69 0.70 0.3129 76.15 0.10468.000 0.002048 0.60 0.59 0.57 0.59 0.2807 85.52 0.13929.525 0.003527 0.53 0.55 0.54 0.54 0.2809 84.77 0.1827
7
Additional info : Oil
Material Weight (N) γ s (N
m3) γ f (N
m3) v (m/s) µ (Ns/m2) V (m3 ¿ r (m)
Mild steel
0.0014 81599.35 8907.48 0.2833 0.1460 1.7157 ×10−8 0.00160.0052 79449.96 8907.48 0.5263 0.8054 6.5450 ×10−8 0.00520.0103 75040.07 8907.48 0.6993 0.2152 1.3726 ×10−7 0.00320.0200 74604.60 8907.48 0.9091 0.2570 2.6808 ×10−7 0.00400.0346 74689.69 8907.48 1.0417 0.3233 4.6325 × 10−7 0.0048
Stainless steel
0.0014 81599.35 8907.48 0.2667 0.1551 1.7157 ×10−8 0.00160.0052 79449.96 8907.48 0.5076 0.8351 6.5450 ×10−8 0.00520.0102 74311.53 8907.48 0.6329 0.2352 1.3726 ×10−7 0.00320.0201 74977.62 8907.48 0.8621 0.2725 2.6808 ×10−7 0.00400.0346 74689.69 8907.48 0.9804 0.3435 4.6325 × 10−7 0.0048
γ s=
ρg/WV
/ Mg
V
v=1mT
V= 43
π r3
8
W =mgAdditional info : Detergent
Material Weight (N) γ s (N
m3) γ f (N
m3) v (m/s) µ (Ns/m2) V (m3 ¿ r (m)
Mild steel
0.0014 81599.35 8613.18 0.9091 0.0457 1.7157 ×10−8 0.00160.0052 79449.96 8613.18 1.3889 0.3065 6.5450 ×10−8 0.00520.0103 75040.07 8613.18 1.6393 0.0922 1.3726 ×10−7 0.00320.0200 74604.60 8613.18 1.9231 0.1220 2.6808 ×10−7 0.00400.0346 74689.69 8613.18 2.0833 0.1624 4.6325 × 10−7 0.0048
Stainless steel
0.0014 81599.35 8613.18 0.9709 0.0428 1.7157 ×10−8 0.00160.0052 79449.96 8613.18 1.2048 0.3533 6.5450 ×10−8 0.00520.0102 74311.53 8613.18 1.4286 0.1046 1.3726 ×10−7 0.00320.0201 74977.62 8613.18 1.6949 0.1392 2.6808 ×10−7 0.00400.0346 74689.69 8613.18 1.8519 0.1827 4.6325 × 10−7 0.0048
γ s=
ρg/WV
/ Mg
V
v=1mT
V= 43
π r3
W =mg
9
5.59 2.97 18.74 25.7 27.870
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Mild Steel Sphere in Oil
Mild Steel Sphere in Oil
Reynold's Number
Drag Coefficient
10
55.45 19.89 99.13 110.72 107.280
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Mild Steel Sphere in Detergent
Mild Steel Sphere in Detergent
Reynold'sNumber
Drag Coefficient
11
4.96 2.76 15.52 22.98 24.680
1
2
3
4
5
6
Stainless Steel Sphere in Oil
Stainless Steel Sphere in Oil
Reynold's Number
Drag Coefficient
12
63.24 14.97 76.15 85.52 84.770
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Stainless Steel Sphere in Detergent
Stainless Steel Sphere in Detergent
Reynold'sNumber
DragCoefficient
13
DISCUSSION
1. Drag force exerted on the solid which move in a fluid is due to viscosity of the fluid. It is made up of two components, surface drag and form drag.
2. Relationship of the drag force with the coefficient is
F=CD ×AρV 2
23. The resultant force on the sphere must equal to the specific weight difference times
the volume of the displaced liquid, strokes expression,
F=4 π r3
3¿
4. The measurement of the drag coefficients of spheres are as follows:
CD=83
r ¿¿
5. Reynolds number
ℜ= ρVDμ
6. The fall velocity of a sphere V is calculated as 1/T, where T is the take time taken by the sphere to fall between the 1m marks. So, in order to calculate the Re, we need to find and determine the viscosity first. Stokes law can be used to find µ as long as Re < 1. The procedure is to select the smallest sphere of the lightest material provided, measure T, calculate V and use this value to find viscosity from following equation.
μ=29
r2¿¿
7. From the above equation we can find the Reynolds number
ℜ= ρVDμ
8. It is possible to measure the value of T for the remaining spheres, and calculate the drag coefficient and the Reynolds number. To get the accurate reading, it is advisable to average the number of T obtained over repeated drops of the sphere.
14
7.0 CONCLUSION
A measurement technique was developed enabling time measurement of spheres falling in
fluids with great accuracy. A proposed mathematical model that includes a new drag
coefficient correction factor enables more precise evaluation of drag coefficients.
15